As outlined in Fig. REF (b), Ch1, Ch2 and Ch0 couple to each other with the same underlying mechanism. For any two channels, for instance, Ch1 and Ch2, the control and probe beams in Ch1 write a collective internal-state coherence into the atomic ensemble, which is then read out by the control beam in Ch2, resulting in a non-Hermitian beam-splitter (BS) type interaction [1]}, [2]} dictated by \(\hat{H}\propto \hat{a}_1^{\dagger }\hat{a}_{2}e^{i\psi _{1}}-\hat{a}_1\hat{a}_{2}^{\dagger }e^{-i\psi _{1}}\) . Here \(\hat{a}_1\)  and \(\hat{a}_2\)  are the annihilation operator of probe beams in Ch1 and Ch2 respectively, and \(\psi _{1}\)  is the relative phase between control beams in Ch1 and Ch2 [3]}.
